Problem: Each chef at "Sushi Emperor" prepares $15$ regular rolls and $20$ vegetarian rolls daily. On Tuesday, each customer ate $2$ regular rolls and $3$ vegetarian rolls. By the end of the day, $4$ regular rolls and $1$ vegetarian roll remained uneaten. How many chefs and how many customers were in "Sushi Emperor" on Tuesday? There were
Solution: Let $x$ represent the number of chefs and let $y$ represent the number of customers. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that each chef prepared $\textit{15}$ regular rolls, each customer ate $\textit{2}$ regular rolls, and $\textit{4}$ regular rolls remained uneaten. How can we model this sentence algebraically? The total number of regular rolls prepared by chefs can be modeled by $15x$, and the total number of regular rolls eaten by customers can be modeled by $2y$. The difference between these quantities is represented by the $4$ regular uneaten rolls, which gives us the following equation: $15 x-2 y=4$ We are also given that each chef prepared $\textit{20}$ vegetarian rolls, each customer ate $\textit{3}$ vegetarian rolls, and $\textit{1}$ vegetarian roll remained uneaten. This can be expressed as: $20 x-3 y=1$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Let's manipulate the equations so one of the variables has the same coefficients but with opposite signs. $\begin{aligned}{-4}\cdot 15x-({-4})\cdot 2y&={-4}\cdot 4\\\\-60x+8y&=-16\end{aligned}$ $ \begin{aligned} {3}\cdot20x-{3}\cdot 3y&={3}\cdot1\\\\60x-9y&=3\end{aligned}$ Now we can eliminate $x$ : − 60 x + 8 y + 60 x − 9 y 0 − y = − 16 = 3 = − 13 \begin{aligned}-60x+8y&=-16\\\\ {+}\ 60x-9y&=3\\ \hline\\ 0-y &=-13 \end{aligned} When we solve the resulting equation, we obtain that $y =13$. Then, we can substitute this into one of the original equations and solve for $x$ to obtain $x=2$. Recall that $x$ denotes the number of chefs and $y$ denotes the number of customers. Therefore, there were $\textit{2}$ chefs and $\textit{13}$ customers.